A systematic study of the problem of adiabatic elimination for systems where inertia as well as multiplicative noise with finite correlation time was taken into account was discussed. It was shown that the limiting equation describing the dynamics in position space depended on the relative magnitude of the two fast-time scales of the system. The multipicative noise in the limiting equation was to be either interpreted in the Itô or the Stratonovich sense depending on whether the noise correlation time tends to zero faster or slower than the particle relaxation time. The results suggest that great care was to be taken in any adiabatic elimination procedure for systems to ensure that the limit was identified.
Bibliographical noteFunding Information:
and to P.R. Kramer for a very careful reading of an earlier version of this paper. G.A.P. and A.M.S. are grateful to EPSRC for financial support. R.K. was supported in part by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities and in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research of the U.S. Department of Energy under Contract DE-AC03-76-SF00098. The authors are grateful to D. Cai and J.C. Mattingly for useful suggestions. They are also grateful to J.M. Sancho for useful suggestions and for providing them with Refs.